An adaptive step-size modified superclass block method for stiff and oscillatory differential systems

Abstract

This paper presents a modified adaptive step-size superclass of block backward differentiation formula (MVSBBDF) for the numerical integration of stiff and oscillatory differential systems. The proposed method extends the two-point variable step-size superclass of block BDF by introducing a positive corrective term that enhances both accuracy and stability. The MVSBBDF computes two solution values simultaneously at each integration step while dynamically adjusting the step size based on local truncation error control. A predictor formula is incorporated to provide starting values for the corrector phase, ensuring robust and efficient convergence. Theoretical analyses establish that the method is of order four, zero-stable, A-stable, and convergent. The step-size selection strategy allows the algorithm to adapt efficiently to stiffness variations within the solution domain. The effectiveness of the proposed method is demonstrated through several stiff and oscillatory test problems. Numerical results confirm that the MVSBBDF delivers improved accuracy, stable convergence, and minimal step rejections while maintaining competitive computational efficiency. Overall, the method provides an effective and robust framework for the numerical solution of stiff systems of ordinary differential equations with adaptive step-size control.